Capacity control in network revenue management:
Clustering and risk-aversion
by
Joongwoo Brian Park
B.S. in Electrical Engineering and Computer Science
Korea Advanced Institute of Science and Technology, 2007
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2010
@
Massachusetts Institute of Technology 2010. All rights reserved.
ARCHfVES
Author ......
Dy
tmet(of Electrical Engineering and Computer Science
December 31, 2009
Certified by. .
...............
Vivek F. Farias
Robert N. Noyce Career Development Assistant Professor in Sloan
School of Management
Thesis Supervisor
A ccepted by ...........
...........
Terry Orlando
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSt UTE
OF TECHNOLOGY
FEB 2 3 2010
LIBRARIES
2
Capacity control in network revenue management:
Clustering and risk-aversion
by
Joongwoo Brian Park
Submitted to the Department of Electrical Engineering and Computer Science
on December 31, 2009, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
Abstract
Network revenue management is the practice of using optimal decision policies to
increase revenues by controlling limited quantities of multiple resources' availability
and prices over finite time. It is widely practiced in capacity-constrained service
industries such as the airlines, hotels, car rentals, and cruise-lines. A variety of
control methods has been introduced for network resource capacity control problem.
We propose a clustering method to improve approximation quality. By clustering the
legs of the network, one can find tighter upperbound than leg-wise decomposition with
loss of computation speed due to larger state space. We have shown that there is more
than 6% revenue improvement opportunity by finding the right clustering. With local
interchange heuristic and generic heuristics, finding a locally optimal clustering can be
done in faster time. We also introduce risk-aversion in network revenue management.
We have investigated risk-aversion on network revenue management and also study
the impact of risk-aversion parameters in the optimization model on relative revenuerisk performance.
Thesis Supervisor: Vivek F. Farias
Title: Robert N. Noyce Career Development Assistant Professor in Sloan School of
Management
4
Acknowledgments
I have made many important choices over the past few years; indeed, one of the best
choice I have made is to work with Professor Vivek Farias. I am so grateful that he
allowed me much freedom in pursuing my own ideas, while giving me enlightening
comments whenever I was lost. His enthusiasm and passion deeply intrigued my
curiosity, and motivated the work in this thesis. I would especially like to thank him
for kindly revising multiple drafts of this thesis and providing prompt and helpful
feedback. Without his guidance and encouragement, this thesis would never have
been possible.
I also thank both my parents Chan Eon Park and Rah Youl Park for endless
support. Without you I could not have finished my thesis.
6
Contents
1 Introduction
2
13
1.1
Clustering in network revenue management . . . . . . . . . . . . . . .
16
1.2
Risk-averse model in network revenue management
17
1.3
Thesis organization............ . . . . . .
. . . . . . . . . .
. . . .
. . . ...
17
General Model
19
2.1
G eneral m odel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.2
Clustering Model..............
. . . . . . . . . . . . ...
21
2.3
Proof..............
. . . . . . . . . . . . . . ...
22
.....
3 Clustering
25
3.1
Clustering impact on revenue
3.2
Heuristics on clustering.......... . . . . . . . . .
3.3
. . . . . . . . . . . . . . . . . . . . . .
30
. . . . . . .
31
3.2.1
Cluster legs that shares the most number of fares . . . . . . .
31
3.2.2
Cluster legs that share high customer arrival rate fare . . . . .
33
3.2.3
Cluster legs that share high priced fare . . . . . . . . . . . . .
34
Local interchange heuristic. . . . . . . . .
. . . . . . . . . . . . .
4 Risk-Averse Model
35
39
4.1
The concept of risk-aversion . . . . . . . . . . . . . . . . . . . . . . .
39
4.2
Single leg risk-averse model
. . . . . . . . . . . . . . . . . . . . . . .
40
4.3
Single leg numerical example . . . . . . . . . . . . . . . . . . . . . . .
42
4.4
Network risk-averse model...
43
. . . . . . . . . . . . . . . . . ...
4.5
5
Network model numerical example
. . . . . . . . . . . . . . . . . . .
Conclusions
5.1
Directions for future research..............
Bibliography
45
51
.. . . ..
. ..
52
53
List of Figures
3-1
(a)A six-node, two-hub airline network.
(b)Example 1:
Two fare
classes at each origin (dashed lines are virtual legs with infinite capacity).
3-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 2: A sixteen-node, product differentiated single hub airline
n etw ork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-3
Histogram of revenue for Example 1 network.
. . . . . .
3-4
Histogram of revenue for Example 2 network.
. . . . . .
3-5
Expected revenue and number of swaps generated by local interchange
heuristic clustering in Example 1 network.......
3-6
Expected revenue and number of swaps generated by local interchange
heuristic clustering in Example 2 network.
3-7
..
. . ...
.
Revenue generated by clustering with local interchange heuristic +
generic heuristic on Example 1.. . . . . . . . . . . . . . .
3-8
Revenue generated by clustering with local interchange heuristic +
generic heuristic on Example 2.
. . . . . . . . . . . . . .
4-1
Network with single leg, LI, with 50 seats available. . . . . . . . . . .
4-2
Expected revenue vs. standard deviation over different k value. . . . .
4-3
Example network for network risk-averse model. Each legs are notated
with leg number and seat capacity in paranthesis.......
4-4
. . . ..
Expected revenue vs. standard deviation over different k value . . . .
42
10
List of Tables
3.1
Price, customer arrival rate, origin-destination pair and path of each
fare product in Example 1 network. . . . . . . . . . . . . . . . . . . .
3.2
27
Price, customer arrival rate, O-D pair and path of each fare product
. . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.3
Cluster matches of Example 1 network for 3.2.1 heuristic. . . . . . . .
32
3.4
Cluster matches of Example 2 network for 3.2.1 heuristic. . . . . . . .
32
3.5
Cluster matches of Example 1 network for 3.2.2 heuristic. . . . . . . .
33
3.6
Cluster matches of Example 2 network for 3.2.2 heuristic. . . . . . . .
34
3.7
Cluster matches of Example 1 network for 3.2.3 heuristic. . . . . . . .
34
3.8
Cluster matches of Example 2 network for 3.2.3 heuristic. . . . . . . .
35
4.1
Expected revenue, standard deviation, and confidence interval of sim-
in Example 2 network.
ulation result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Expected revenue, standard deviation, and confidence interval of simulation result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
46
Expected revenue, standard deviation, and confidence interval of simulation result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
42
47
Expected revenue, standard deviation, and confidence interval of simulation result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
12
Chapter 1
Introduction
Network revenue management is the practice of using optimal decision policies to
increase revenues by controlling limited quantities of multiple resources' availability
and prices over finite time.
It is widely practiced in capacity-constrained service
industries such as the airlines, hotels, car rentals, and cruise-lines. Historically, revenue management originates from the Airline Deregulation Act of 1978. With this
act, the U.S. Civil Aviation Board loosened control of airline prices, which had been
strictly regulated based on standardized price and profitability targets.
Since the
enormous potential profitability of revenue management was recognized due to the
deregulation, mathematical models have been developed to determine sophisticated
capacity control strategies. Current methodology can be divided into leg-based and
network-based models.
Leg-based methods are aimed at optimizing the passenger
mix on a single-leg flight. Network-based models consider booking requests for multiple legs simultaneously. Developments in the field of leg-based models started with
the study of a single-leg example by Littlewood
[7] with two fare classes under the
assumption that low-fare class passengers book first. Richter [8] shows that Littlewood's intuitive rule for when to close down the second fare class in fact is optimal.
Belobaba [2] generalizes this approach to a heuristic strategy for the case of multiple
fare classes, which is widely used in practice under the term expected marginal seat
revenue (EMSR). Wollmer [13] shows that the assumption of a strict arrival order of
demand for different fare classes allows optimal booking control in the form of static
booking limits and proposes algorithms to calculate these. A major flaw of leg-based
models is that they only locally optimize booking control, whereas an airline should
strive to maximize revenue from its network as a whole. These objectives might even
conflict. To see this, think of a booking request for a high-fare class flight from Singapore to Amsterdam. Alternatively, this seat can be sold to a passenger of a lower fare
class traveling from Singapore to New York via Amsterdam. When the direct flight
from Amsterdam to New York has ample capacity, granting the latter request may
be more profitable if it has a higher ticket price. But if booking control is optimized
locally, the former request should be granted, because that is the one for the highest
fare class.
Airlines typically offer hundreds of such combinations of origin, destination and
fare class. Hence taking a network approach is more realistic and can result in significant improvement in revenue. Simulation studies of network methods by various
researchers have demonstrated notable revenue benefits from using them over singleresource methods to airline networks. [12][4][6] However, determining an overall booking control strategy for the entire network is far from trivial. The scale of the problem
thus prohibits the use of realistic dynamic programming models as in the single-leg
case. For example, any realistic network with 30 resources and capacities of 100 on
each resource has 10030 states to compute at each stage. Hence research has concentrated on developing tractable heuristics instead. Achieving a good balance between
the quality of the approximation and the efficiency of the resulting algorithms becomes the primary challenge.
A variety of control methods has been introduced for network resource capacity
control problem. Booking limits is one type of control for network revenue management where it limits the amount of capacity that can be sold to any particular class
at a given point in time.
For instance, a booking limit of 19 on class 2 indicates
that at most 19 units of capacity can be sold to customers in class 2. Beyond this
limit, the class would be closed to additional class 2 customers. This limit of 19 may
be less than the physical capcity, but can protect capacity for future demand from
class 1 customers. Booking limits can be either partitioned or nested. Partitioned
booking-limit allocate a fixed amount of capacity on each resource for every product
that is offered. These allocated amounts of capacity do not overlap; demand for a
product has exclusive access to its allocated capacity, and no other product may use
this capacity. For example, with 30 units to sell, a partitioned booking limit may set
a booking limit of 12 units for class 1, 10 units for class 2, and 8 units for class 3.
If the 12 units of class 1 capacity are used up, class 1 would be closed regardless of
how much remaining capacity is available. This could be undesirable if class 1 has
higher revenues than do classes 2 and 3 and the units allocated to class 1 are sold
out but there still exists demand for class 1. This static fragmenting of capacity can
result in tremendous inefficiencies when demand is stochastic. With a nested booking
limit, the capacity available to different classes overlaps where higher-ranked classes
having access to all the capacity reserved for lower-ranked classes.[10] If the nested
booking limit for class i be denoted bi, then bi is the maximum number of units of
capacity that can be sold to classes i and lower. For instance, if b1 = 20, b2 = 11,
and b3 = 6, then at most 20 bookings for classes 1,2, and 3 can be accepted, at most
11 for classes 2 and 3 combined , and at most 6 for class 3 customers. This simply
allows any remaining capacity after selling to low classes to become available for sale
to higher classes. However, it is difficult to specify booking limits for products that
are consistent across the resources in the network, since multiple products share the
multiple resources. One other type of control is bid-price control that sets a threshold
price for each resource in the network[9]
[5].
The bid price is normally interpreted as
an estimate of the marginal cost to the network of consuming the next incremental
unit of the resource's capacity. When a request for a product comes in, the revenue
of the request is compared with the sum of the bid prices of the resources required
by the product. If the revenue exceeds the sum of the bid prices, the request is
accepted; if not, it is rejected. One strategy for generating bid-price network controls is to decompose the rn-resource problem into m single-resource problems, each
of which may incorporate some network information but are solved essentially independently. Formally one can think of such a decomposition method as follows. An
approximation method decomposes the network problem into m single-resource prob-
lems and applies a single-reource method Mi on each resource i, with value functions
VMt (xi), that depends on the time to go t and the remaining capacity xi of resource
i. Where the value function measures the optimal expected revenue as a function
of remaning capacity xi. These may be constructed by incorporating some static,
network information into the estimates. The the total value function is approximated
by Vt(x) = > Vmj(xi). Origin-destination factors method by Belobaba [3] is a simple
i=1
type of decomposition approximation where one simply takes single-resource methods that one might already have in place and convert their outputs into estimates
of network displacement costs. Willamson [12] showed prorated expected marginal
seat revenue (PEMSR) sheme that involve allocating a portion of the revenue of each
product to the resources used by the product. Naturally generalizing, clustering number of single-resource in to a cluster to approximate the network problem has been
introduced.
1.1
Clustering in network revenue management
In this paper, we propose a clustering method to improve approximation quality. In
this method, resources are clustered into a specified number of classes with equal
number. For instance, m-resource network that has been decomposed into m single
resource problem can be grouped to c number of classes with equal
[M]
number re-
sources in each cluster. In this resource clustering method, control strategy proceeds
as following. A request for a product is converted into requests for the corresponding
resources on each class required by the product. If the resources on each class are
available, the request is accepted.
If the one or more resources on each class are
closed, the request is rejected. In this scheme, larger number of resources per class
improve the quality of approximation. We have investigated the importance of clustering based decomposition methods for network revenue management. By capturing
gap of revenues generated by different clusterings, we can calculate the impact of
clustering on total revenue. Also, generic clustering heuristics are studied based on
different clustering criteria that are based on customer demand rate, product price,
and co-dependency among the resources. Moreover, a local interchange heuristic is
introduced to find a locally optimal clustering.
1.2
Risk-averse model in network revenue management
In the fouth chapter of this thesis, we introduce risk-aversion in network revenue
management.
Despite the very rich literature on revenue management, there are
only very few papers that deal with risk-aversion in the broader context of revenue
managememnt, and even less that maximize the expected utility of the decision maker.
Weatherford [11] first used risk-aversion in a capacity control model. In his approach,
a product's revenue is replaced by the utility of its revenue that the heuristics can have
significant impact on the expected uitlity and revenue performance and increases the
probability of hitting certain revenue theresholds. Barz [1] extended Weatherford's
simulation based risk-aversion heuristic into more generalized single-resource capacity
control problems from the perspective of a risk-averse decision-maker. She suggested
more realistic optimal controls of an expected atemporal utility maximizing policy
with constant absolute risk-aversion. However, there are no other papers that treat
risk-aversion on network revenue management. We will investigate risk-aversion on
network revenue management and also study the impact of risk-aversion parameters
in the optimization model on relative revenue-risk performance.
1.3
Thesis organization
The remainder of the thesis is organized as follows. In Chapter 2, we first introduce
general problem description and models. Also, basic concepts and terminology will
be explained. In Chapter 3, importance of clustering will be shown with numerical
examples in two different networks. In Chapter 4, risk-averse model on single leg and
multi-leg network will be introduced with numerical simulation examples. In Chapter
5, conclusions with future research direction will be shown.
18
Chapter 2
General Model
2.1
General model
We consider a dyanamic, stochastic model of revenue management problems that
addresses the joint allocation decision in a multiple-product, multiple-leg network
f, f
setting.
To provide a unit of product
I E £C
{1, ..., L}. We define the matrix A = [Aij ], where A, 1 E {0, 1} depending
on whether fare
f
=1,
... ,
F requires A 1,f units of leg 1,
consumes a seat on leg 1 or not. Each fare product is associated
with a price Pf and requires seats on one or more legs. Initial capacity on each leg
is given by a vector xO E Z . Time is discrete. We assume a T period horizon with
at most one customer arrival in a single period. A customer for fare product
f arrive
in the t-th period with probability Af. We note that the discrete time arrival process
model we have described may be viewed as a uniformization of an appropriately
defined continuous time arrival process. At the start of the t-th period the airline
must decide which subset of fare products from the set
f
sale; an arriving customer for fare product
{f
: Af < Xt} it will offer for
is assigned that fare product should it
be available, the airline receives Pf, and xt+1 = xt - Af.
We define the state-space S =
{
:
E
,
xo}x {t:t EZ+,t<T} and
encoding the products offered for sale at time t by a vector in a where a E
{0,
1I}F.
State transition function, S, is mapping S x {0,1}F 4 S. It can be described as
below transition function.
(x - A1 , t + 1), w.p. Aia1I{fx>Aj
(2.1)
,if t < T
S(x, t, a) =
(x, t + 1), w.p. 1
(
-
AfaflIxAf1)
f=1
(X, t)
A control policy is a mapping 7r : S
,
{O,1}F.
24
otherwise
Let IT be the set of all such
policies. Let R be a random variable representing revenue generated depending on
which state we are in. Then
(2.2)
P1 w.p.
A1a1I{2 A 1}
P2 w.p.
A2a2I{xA
2}
R(x, t, a)
PF w.p.
0 w.p. 1 -
AFaF{x AF}
F
(E
f=1
And we let maxGEJ'(xO,0)
AFaFI{x AF})-
= J*(xo,0), denote the expected revenue under the
optimal policy 7r* upon starting in state (x, t). J'(x, t) is defined as
~T-1
(2.3)
J'(xo, 0) = E
R(Xt, 7r(Xt))
jXo = xo
..t=0
where Xt = S(Xt_ 1, 7r(Xt_ 1 )).
J* and 7r* can be computed via dyanmic programming. We can define the dynamic
programming operator T according to
(TJ)(,t-1)= 1-S)Af
J(x,t)
F
(2.4)
+
5 max (AJIx;>Af}I (Pf + J (x - Af, t)) , Af J (x, t))
f=1
af
We define (TJ)(x, T) = 0 for the T-th horizon. J* can then be idenified as the unique
solution to the fixed point equation TJ = J. -r* is the greedy policy with respect to
J*.
2.2
Clustering Model
We propose a clustering model that decompose the original network into clusters.
Assume we are given with clusters C, C {1, 2, ..., L} where c = 1, . . . , C indicate the
n Cj = #,
index of cluster C. We assume Ci
one cluster, and
U Ci
=
i.e. each leg can only be associated with
{1, 2,. .. , L}, i.e. clusters are collectively exhaustive. For
convenience, we assume IC
= C.
Assume that Cc is an ordered set
{/i, 12,.. - , lle}. Let
vector on cluster Cc. x' is defined so that
xC
cEZL"
xC be the seat availability
with x = xz
where the seat
availability of ith leg in cluster C is the seat availability of leg 1i in original system.
Adapting notations from the general model described in section 2.1, we define the
matrix AC = [Acf] where Ac
= AiIlecc. Fare coefficient for each cluster is defined
L
as a
Z
Acf
Z
AIf
=
.
=1
We define the state-space Sc
{
:E
Z
<XC} x
{t :t E Z
< T} and
encoding the products offered for sale at time t by a vector in a where a E
For each cluster, state transition function, SC, is mapping Sc x
{0,
{0, 1}F
1}F.
SC
can be described as below transition function.
(Xc -
(2.5)
Ae, t + 1), w.p.
Aia1IJ{c>A-
if t < T
S(XC, t, a) =
F
(XCt
+ 1), w.p. 1-
(1
f=1
(XC, t)
A control policy is a mapping
7tc
: SC
4
AfafI{xc Ac I)
J
,
otherwise
{f, 1}F. Let Ic be the set of all such
policies. Let Rc be a random variable representing revenue generated by the network
that is described below.
arP1 w.p. Aia1I{Xc>Al}
aoP 2 w.p.
A2a2J{xCe>Al}
R(xc, t, a) =
(2.6)
acFPF w.p. AFaF{c>Ac}
F
F
F
(E
0 w.p. 1 -
AFaF{xc;>A})
f=1
And we let max JcTc (xo, 0) = J* (xo, 0), denote the expected revenue under the optimal
IrcEne
policy 7r* upon starting in state (XC, t). Jerc(xc, t) is defined as
~T-1
Jlc(xc, 0) = E
(2.7)
[
R(X 7C(X)) XC = zo
.
t=o
where Xt = Sc(X_1, 7rc(Xt_1)).
J* and
7* can be computed via dyanmic programming. We can define the dynamic
programming operator T according to
(T Jc)(xc, t - 1) =
-
A\f
Jc (z", t)
(2.8)
+Znmax (AfIxc;>Ac} (Pf ac + Jc
f=1
(xc
-
A?,t)) , Af Jc (xc, t))
We define (TJc)(xc, T) = 0 for the T-th period. J* can then be idenified as the unique
solution to the fixed point equation TJc = Jc. 7r* is the greedy policy with respect to
Jc*- Notice that if the sizes of the clusters are small, computing J* is easy.
2.3
Proof
C
We can show that Z J*(XC, t) is an upper bound of J*(x, t). In particular we show
C=1
C
Proposition 2.3.1.
E
C=
1
J*(XC, t) > J*(X, t) V(x, t) E S.
Proof. First, we want to prove in case for (T - 1)th period, where
C
SJ*(X, T - 1) > J*(x, T - 1).
c=1
Left-hand side equation can be expanded as below.
F
Afj{xc>A}
aPf
J*(XC,T - 1) =3
f=1
Right-hand side equation can be expanded as below.
F
J*(x,T - 1)
Pf Af I{
-
>A,}
f=1
Inequality can be proved as below.
C
F
C
fEf Ixc>A}
J'(XC*,T - 1) =CF
c-1
c=1 f=1
C
+} - '
11(cPAiJ{x>Ac
'
aFPFAFI{x;>A})
c=1
C
C
1cPiA
1 {zc;>Ac}
C
c PFAFF{xc>Ar}
''-
+
c=1
1A1J{>A1}
*. *>3
Y4PFAFF{x>AF}
c=1
c=1
L
L
c
EA, 1
Y
c=1
C
PA1I{fx>A1}
Aj
+
-.
±>3-'y+FAF{zx>AF}
c=1
1=1i
PiAi{z>A1} +
F
>3Pf AfIfx;>Af}
f=1
- J*(xT - 1)
ZA1,F
E Al,F
1=1
'
+
FAFIJ{x>AF}
For (T - k)th period, we want to prove that:
C
(z', T - k) > J*(x, T - k)
),J*
c=1
{f : 7r*(x, t - k) = 1} and Fo
First, we can define F1
{f : 7r*(x, t - k) = 0}.
Then,
C
C
E
c=1
C
+
F
1-
=E
c=1
J*(z',IT -k)
E Af
J* (xc,T-
f=1
(k+ 1))
F
E max(Af (I{2c
f-1 af
c
c-1
C
(
> E
c=1
AI(Pfac + Jc*(x - Ac, T - (k + 1))), J*(xc,T - (k + 1))))
F
1-
ElAf
J* (X',T -(k
f=1
+1))
C
+
C
A1)))+
f CFi
c=1
f EFo
C
-
E
)
Af [Pf EC oz+
1I- E Af
f=1
T
-
(k+
1))
C
EJ*(xc - A
c=1
c=1
f E F1
+
J*(X,
Af
f=1
1
+ E
>
c=1
F
E
c=
E AfE(Jc*(xc,t-(k+1)))
+
C
,
-
(k + 1))] +
E Af E(J*(xc,t - (k + 1)))
f EFo
c=1
J*(x, T - (k + 1))
E (Af Pf + J*(x- Af,T - (k+ 1))) + E Af J*(x, T - (k + 1))
f E F1
fE Fo
=
E Af
J*(x, T - (k + 1))
f=1
F
+
E
f=1
max(A(
af
1
--
(
J*(x,T - k)
Due to the induction theorem, this inequality holds for all period.
Chapter 3
Clustering
Clustering legs can be important for value function approximations based on decomposition in network revenue management.
We next examine two exemplary cases
of our approach and the results of some numerical experiments to find out the importance of clustering (i.e. selecting appropriate sets, Cc, for the approximations
described in section 2.2).
2
6.
+
2
(150
L3
(50)
L9
(150)
L5
L2 (1 6
L4
(200)1
(100)
L7
4
(150)
L8
L10
(50)
(a)
L9
(150)
L6 L4
.
L2
.. '
(100)
(100)
(100)
(10
L1
3
(50)
,
L11
(50)
L11
(50)
0)
L5
200)
(50)
3
L3
s50)
5
.
5
(b)
Figure 3-1: (a)A six-node, two-hub airline network. (b)Example 1: Two fare classes
at each origin (dashed lines are virtual legs with infinite capacity).
We have adapted Esample 1 network from Gallego and van Ryzin's paper [6].
Figure 3-1(a) shows a network problem that has no product differentiation with two
"hub" nodes at node 2 and 3. Leg seat capacities are shown in parenthesis and were
chosen to approximate the number of seats on a single aircraft. We then extended
the example network of Figure 3-1(a) to more realistic example network of Figure
3-1(b). In this network, suppose a well differentiated super-saver and full-coach product exists for each origin-destination pair of the network from Figure 3-1(a). These
products are differentiated by travel restrictions, cancellation policies or other mechanisms not related to the time of purchase. In this way, sales of the two products
occur concurrently throughout the time horizon. We model product differentiation
using virtual nodes at each city I to represent the demand from each market segment.
These virtual nodes are then connected to the physical node I via infinite capacity
links. Thus, the virtual nodes compete for the same physical capacities on the legs of
the network. Figure 3-1(b) shows the network of Figure 3-1(a) modified using virtual
nodes to account for two classes of demand originating at each node.
Parameter values for all fare products are shown in Table 1 along with the path
(itinerary) used by each fare. The values shown in Table 3.1 are essentially arbitrary
and were chosen merely to illustrate the performance of the heuristics; however, it is
not hard to see that the revenue of the heuristics is affected only by the deterministic fare prices and customer arrival rate. These prices and customer arrival rates
are reasonable approximations of those found in actual airline applications as of the
writing of Gallego and van Ryzin [6].
Fare Number
Price
Customer Arrival Rate
Origin
Destination
Path
1
749.17
0.00454421
7
2
7->1->2
2
804.34
0.004168894
7
3
7->1->3
3
1001.79
0.004187879
7
4
7->1->2->4
4
866.25
0.003457971
7
5
7->1->6
5
879.32
0.004171437
7
6
7->1->3->5
6
160.89
0.05825701
8
2
8->1->2
7
216.85
0.020401622
8
3
8->1->3
8
266.44
0.021866416
8
4
8->1->2->4
9
224.58
0.013329131
8
5
8->1->6
10
328.55
0.004259988
8
6
8->1->3->5
11
775.23
0.004526011
9
3
9->2->3
12
692.18
0.004621867
9
4
9->2->4
13
830.13
0.003672534
9
5
9->2->3->5
14
740.35
0.004265312
9
6
9->2->4->6
15
160.92
0.04844433
10
3
10->2->3
16
135.04
0.062987514
10
4
10->2->4
17
271.67
0.002598419
10
5
10->2->3->5
18
183.21
0.020470194
10
6
10->2->4->6
19
674.87
0.00475715
11
2
11->3->2
20
681.95
0.004701345
11
4
11->3->4
21
615.04
0.004281708
11
5
11->3->5
22
429.95
0.003494762
11
6
11->3->4->6
23
124.87
0.048960054
12
2
12->3->2
24
131.64
0.042760071
12
4
12->3->4
25
156.71
0.013837257
12
5
12->3->5
26
154.8
0.00121702
12
6
12->3->4->6
27
348.17
0.00611991
13
6
13->4->6
28
73.88
0.049895615
14
6
14->4->6
15->5->3->2
29
838.75
0.00362015
15
2
30
663.88
0.003883248
15
3
15->5->3
31
765.79
0.003359388
15
4
15->5->3->4
32
753.93
0.002844689
15
6
15->5->3->4
33
288.75
0.001846709
16
2
16->5->3->2
34
222.21
0.01318514
16
3
16->5->3
35
288.85
0.000757778
16
4
16->5->3->4
36
298.45
0.000247277
16
6
16->5->3->4->6
Table 3.1: Price, customer arrival rate, origin-destination pair and path of each fare
product in Example 1 network.
2
414'
4
L9
(150)
L1
L8
(150)
(50)
*.
OsWL
(50)
(50).*
L6
5
...
1
L4
.*(100)
(100)
(50)
L
(100)
3
11 +
Figure 3-2:
""""
L5
(200)
1
(50)
6
Example 2: A sixteen-node, product differentiated single hub airline
network.
We also consider another exemplary product differentiated network with illegs
and 36 fare products depicted in Figure 3-2. In this example, main hub is located at
node 3, and two regional hubs are at node 1 and 5. Node 2 and 4 plays as connector
and node 6 is arrival only destination. Parameter values for example 2 network is
described in Table 3.2.
Fare Number
Price
Customer Arrival Rate
Origin
Destination
Path
1
749.17
0.00454421
7
2
2
804.34
0.004168894
7
3
7->1->2
7->1->3
3
1001.79
0.004187879
7
4
7->]->2->3->4
4
866.25
0.003457971
7
6
7->1->3->6
5
879.32
0.004171437
7
5
7->1->3->5
6
160.89
0.05825701
8
2
8->1->2
7
216.85
0.020401622
8
3
8->1->3
8
266.44
0.021866416
8
4
8->1->2->3->4
9
224.58
0.013329131
8
6
8->]->3->6
10
328.55
0.004259988
8
5
8->1->3->5
11
775.23
0.004526011
9
3
9->2->3
12
692.18
0.004621867
9
4
9->2->3->4
13
830.13
0.003672534
9
5
9->2->3->5
14
740.35
0.004265312
9
6
9->2->3->6
15
160.92
0.04844433
10
3
10->2->3
16
135.04
0.062987514
10
4
10->2->3->4
17
271.67
0.002598419
10
5
10->2->3->5
18
183.21
0.020470194
10
6
10->2->3->6
19
674.87
0.00475715
11
1
11->3->1
20
681.95
0.004701345
11
4
11->3->4
21
615.04
0.004281708
11
6
11->3->6
22
429.95
0.003494762
11
5
11->3->5
23
124.87
0.048960054
12
1
12->3->1
24
131.64
0.042760071
12
4
12->3->4
25
156.71
0.013837257
12
6
12->3->6
26
154.8
0.00121702
12
5
12->3->5
27
348.17
0.00611991
13
3
13->4->3
28
73.88
0.049895615
14
3
14->4->3
29
838.75
0.00362015
15
4
15->5->4
30
663.88
0.003883248
15
3
15->5->3
31
765.79
0.003359388
15
6
15->5->6
32
753.93
0.002844689
15
1
15->5->3->1
33
288.75
0.001846709
16
4
16->5->4
34
222.21
0.01318514
16
3
16->5->3
35
288.85
0.000757778
16
6
16->5->6
36
298.45
0.000247277
16
1
16->5->3->1
Table 3.2: Price, customer arrival rate, O-D pair and path of each fare product in
Example 2 network.
3.1
Clustering impact on revenue
In order to understand significance of clustering on revenue, we have simulated 120
times on Example 1 network, by clustering random pairs of legs together. Also,
simulated 70 times on Example 2 network by clustering random pairs of legs together. Maximum revenue generated was 105475.5 and minimum was 99321.47 with
standard deviation of 1510.612 as for Example 1 network. For Example 2 network,
maximum revenue was 106534.6 and minimum was 98743.93 where standard deviation was 1613.006. The point of these experiments is to show the careful clustering
impact. Where it can have on the performance of the resulting policy - upto 6.6% in
the examples above. In all cases, the policy used is the greedy policy with respect to
the approximation
E J*.
The middle 90% of revenue generated is within 101000 to
C
105000 where distribution is centered at 102800 as depicted in Figure 3-3. Histogram
on both cases shows that it is median centered distribution.
16
14
12
A
10
--------------
8
6
4
98000
100000
102000
104000
106000
Expected Revenue
Figure 3-3: Histogram of revenue for Example 1 network.
10
8
-- -----------
6
4
2
0
98000
100000
102000
104000
106000
108000
Expected Revenue
Figure 3-4: Histogram of revenue for Example 2 network.
3.2
Heuristics on clustering
Although price improvement opportunity can be more than 6% by finding the right
clustering, it can take very long time to find the optimal clustering. For example, in
order to find best two leg clustering of 11 leg network, it requires 1247400 (approx.
1.2million) number of simulation to search the best clustering. We consider several
generic heuristics to find clustering that maximize the revenue.
3.2.1
Cluster legs that shares the most number of fares
The decomposition approximation can apparently be improved by clustering legs that
share the most number of fares in common. By clustering legs that share most number
of fares in Example 1 network, revenue is 102992.7712 where average of random
clustering is 102256 and minimum is 100826 for two leg clusters. This heuristic gave
the revenue improvement of 2.11%. The clusters used in our experiment for Example
1 and Example 2 network are given in Table 3.3 and Table 3.4. In particular, we used
the following procedure to cluster legs.
1. Randomly select unclustered leg.
2. Select an unclustered leg that share as many fares as possible with the leg picked
in step 1.
3. Put legs selected from step 1 and step 2 in the same cluster.
4. Go back to step 1 until there is no unclustered leg to select.
Cluster
Component Component
Sharing
1
1
Li
2
L5
Fares
3,8
2
L4
L8
13,17
3
L7
L9
22,26,36
4
L2
L3
Null
5
L6
L11
33,29
6
LIO
Null
Null
Table 3.3: Cluster matches of Example 1 network for 3.2.1 heuristic.
Clustei Component
1
Li
1
Component
2
L9
Sharing
Fares
3,8
2
L3
L5
4,9
3
L4
L6
32,36
4
L2
L3
Null
5
L6
11
Null
6
LIO
Null
Null
Table 3.4: Cluster matches of Example 2 network for 3.2.1 heuristic.
3.2.2
Cluster legs that share high customer arrival rate fare
It is natural to expect that the quality of our approximation depends not only on
the number of fares shared among legs in a cluster, but also, the arrival rate for
these fares. So we can consider a heuristic that clusters legs that share high customer
arrival rate fare. In particular we cluster legs as following:
1. Rank fares by arrival rate.
2. Cluster legs required for the highest ranked fare.
3. Remove top ranked fare from the consideration and repeat step 2 for unclustered
legs.
The revenue was 103598 with price improvement of 2.71% for Example 1 network.
For example, since fare 8 has the highest arrival rate among the fares that require
multiple legs, leg 1 and 5 can be clustered together. The clustering used are given in
the table below.
Cluster
Component
Component
1
2
1
LI
L5
2
L4
L7
3
L6
L9
4
L2
L8
5
L3
L11
6
L1O
Null
Table 3.5: Cluster matches of Example 1 network for 3.2.2 heuristic.
Cluster
Component
Component
12
1
Li
L9
2
L4
L5
3
L6
L8
4
L2
L7
5
L3
L11
6
LIO
Null
Table 3.6: Cluster matches of Example 2 network for 3.2.2 heuristic.
3.2.3
Cluster legs that share high priced fare
We can also cluster the legs based on the price rank of the fares. It generated revenue
of 102960 with price improvement of 2.08%. In our example network, fare 3 has
the highest price that for this heuristic we can cluster leg 1 and 5 together. Next
expensive fare is fare 5 which demands leg 2 and 11. So we can cluster leg 2 and 11
into one cluster. The procedure used exactly as in section 3.2.2, with the exception
that fares were ranked by fare prices.
Component
Component
1
2
1
Li
L5
2
L2
L11
3
L6
L8
4
L4
L7
5
L3
L9
6
L10
Null
Cluster
Table 3.7: Cluster matches of Example I network for 3.2.3 heuristic.
Component
Component
1
LI
L2
2
L4
L7
3
L3
L9
4
L5
L8
5
L6
L11
6
LIO
Null
Cluster
Table 3.8: Cluster matches of Example 2 network for 3.2.3 heuristic.
3.3
Local interchange heuristic
We can attempt to improve upon the clusterings produced by any of the aforementioned heuristics by employing the following procedure.
1. Start with a random clustering (or a clustering produced from one of the previous heuristics).
2. Consider an arbitrary pair of clusters; say (i, j) and (k, 1).
3. Consider all swaps of legs between the two chosen clusters, (i, k) and
(j, 1),
or
(i, l) and (j, k).
4. Choose the swap that maximize the simulated revenue.
5. Go back to step 2.
6. Stop when no pair of clusters yields an improvement in step 3.
With this scheme, we have ran 10 simulations with local interchange heuristic on
both Example 1 and Example 2 networks. For Example network 1, this generated
revenue that is higher than other generic heuristics with average revenue of 104965.256
and maximum 105780.4. In the best case, it improved revenue by 4.84% by swapping
max 40 times for Example 1 network. Since its local heuristic, it generates different
revenue depending on the starting sequence of clusters.
110000
100000
90000
80000
70000
60000
50000
1
2 131415
6
7
8
9
10
105301105731 104053 104936 105779 104457 105391104737
ce0eExpectedRevenue 104645 014622
22
17
23
40
32
16
21
31
25
8
j@NumberofSwaps
J
Figure 3-5: Expected revenue and number of swaps generated by local interchange
heuristic clustering in Example 1 network.
110000
100000
90000
80000
60000
70000
50000
ExpetedRevenue
* j=Nmerofwps
*
9
8
7
6
5
4
3
2
103767 103689 04314 104167 03219 0353 104866104144 1038591104572
26
30
16
50
22
38
32
16
19
45
Figure 3-6: Expected revenue and number of swaps generated by local interchange
heuristic clustering in Example 2 network.
Benchmarking from generic heuristic as starting point, we can use local inter-
change heuristic to improve return revenue. With generic heuristic from section 3.2.2
as starting point, clustering with local interchange heuristic returned 107276.4 as revenue. This benchmark gave the best performance with improving revenue by 6.3%.
120000
110000
100000
90000
80000
70000
60000
50000
2
3
Expected Revenue
105732
107276.4
105301.1766
Numberofwaps
18
25
31
Figure 3-7: Revenue generated by clustering with local interchange heuristic + generic
heuristic on Example 1.
120000
110000
100000
90000
80000
70000
60000
50000
40000
30000
20000 0
flExpectedfRevenue
- -
2----3-
105358.6556
05747.622
105447.6947
26
30
16
Figure 3-8: Revenue generated by clustering with local interchange heuristic + generic
heuristic on Example 2.
38
Chapter 4
Risk-Averse Model
4.1
The concept of risk-aversion
Decision makers might differ in their attitude towards risk. Within the theory of
expected utility, different attitudes towards risk can be expressed by the shape of the
utility function.
An expected utility maximizing decision-maker with utility func-
tion u is called risk-averse if one prefers the expected outcome of a non-degenerate
lottery to the lottery itself. Representing a lottery by the random variable W with
expectation E[W], this is equivalent to
(4.1)
u (E[WJ) > E[u(W)]
for all lotteries W. Jensen's inequality yields that a decision-maker with utility function u is risk-averse if and only if the utility function u is concave. In case of equality
in (4.1), decision-maker's attitude is called risk-neutral. This holds for linear utility
functions. Given the reverse inequality, the decision maker is risk-loving, which corresponds to a convex utility function.
Now suppose the decision-maker had the choice between playing the lottery with
random outcome W or receiving a certain amount of money w. The certainty equivalent of a lottery is the value of w that makes the decision-maker indifferent between
the two options. Thus, it fulfills
E[u(W)] = u(w).
(4.2)
Consequently, the certainty equivalent can be defined as
w =u
(4.3)
If the utility function u is invertible.
1
(E[u(W)]).
Choosing the lottery with highest expected
utility corresponds to choosing the lottery with highest certainty equivalent.
If we assume that the decision-maker is risk-averse, he prefers the expected outcome of a lottery to playing the lottery. In other words,
u(E[W]) > E[u(W)]
(4.4)
=
u(w),
using (4.1). Consequently, w < E[W] owing to the assumed monotonicity of u.
4.2
Single leg risk-averse model
Extending from general model described in Chapter 2, we now consider a model where
decision maker is in risk-averse attitude.
In risk-averse model, objective changes
from maximizing revenue to maximizing utility. Utility can be described from utility
function u(w) that is nonlinear and takes accumulated wealth, w, as variable. In order
to solve this sequential decision problem that aim at maximizing expected utility can
be solved by a Markov decision process if the state space is enlarged by another
variable w, the accumulated wealth up to the current period.
For single leg problem we define the state-space1 S =
w E Z+, w '
{x :
E Z+, x < xo}
x
{w
wo} x {t : t E Z+, t < T} and encoding the products offered for sale
at time t by a vector in a where a E {0, 1}F = A. State transition function, S is
'We assume prices are integer valued.
mapping S x {0, 1}F
S4
S. It can be described as below transition function.
(4.5)
(X - 1, w + Pi, t + 1), w.p. Aia1I{xo}
if 0 < t < T
a,
S
(X,w, t + 1), w.p. 1 -(F
AfafI{f>o)
,
(X, w, t)
otherwise
A control policy 7r is mapping function 7r : S 1+ {0, 1}F. Let I be the set of all
such policies. Let R(x, w, t, a) be a random variable representing revenue generated
by the airline when fare products a E A are offered for sale, and we have x seats left.
Then,
T-1
R(Xt, 7r(Xt)) Xo =o
J'(xo, wo, 0) = E
(4.6)
.t=oI
where Xt = S(Xt_ 1 , r(Xt_1)). Then let maxrnJ'(x,w, t)
J*(x, w, t), denote the
expected revenue under the optimal policy 7r* upon starting in state (x, w, t) E S.
J* and r* can be computed via dynamic programming. We can define the dynamic
programming operator T according to
(T J) (x, FF
w, t)
(4-7)
+
=
1 - E Af
f=1
J(z, w, t + 1)
F
E Af max (If.,>o (J(x - 1, w + Pf, t + 1), J(x, w, t + 1))
f=1
af
+ (1 - I(>oj) J(x, w, t + 1))
We define (TJ)(x, w, T-1) = u(w) for the (T -1)-th horizon. J* can then be idenified
as the unique solution to the fixed point equation TJ = J. 7r* is then the policy that
achieves the maximum utility in (4.7).
J(x, w, t + 1) and t < T.
So, r* = 0 iff J(x - 1, w + Pf, t + 1) <
4.3
Single leg numerical example
We next examine numerical example of our model. In this example, there is 50 seats
available for a single leg route, where finite time horizon of 1000 steps. This route
starts from origin A and ends at destination B with only one leg as depicted in Figure
4-1.
There are two fare products offered in this route where arrival process is un-
correlated to each other. Fare 1 has time homogeneous arrival with customer arrival
rate of 0.1. Fare 2 also has constant customer arrival rate of 0.05 where customer
only arrives after 500th time step. Fare 1 is priced at 100 and Fare 2 is priced at 200.
(50)
Figure 4-1: Network with single leg, L1, with 50 seats available.
With given setting, we can maximize expected utility by computing via dynamic
programming model from section 4.2. u(w) = w(1-k) is used for risk-averse utility
function, where k is risk-aversion parameter. So, more risk-averse it is, higher the k
value.
With different risk-aversion k values, we computed value functions via dynamic
programming and ran simulation from stored value function to calculate expected
revenues and standard deviations of these revenues. Simulation results shows that As
risk-aversion parameter k increases, standard deviation decreases.
Risk Aversion
Expected
Standard
Parameter k
Revenue
Deviation
0
0.9
1-10-9
1-10' 5
7432.54
7421.08
6787.4
5088.66
483.1499
480.1393
354.5888
148.4918
Confidence
Range
Minimum
7431.012146
7419.561666
6786.278692
5088.190428
Interval
Range
Maximum
7434.067854
7422.598334
6788.521308
5089.129572
Table 4.1: Expected revenue, standard deviation, and confidence interval of simulation result.
8000
7000
k k=0.9
6000
,
S5000
k=1-10
.~4000
3000
rl 2000
1000
0
0
200
100
300
400
500
600
Standard Deviation
Figure 4-2: Expected revenue vs. standard deviation over different k value.
As shown in Figure 4-2, as risk-averse parameter k increases, standard deviation
decrease and also expected revenue decrease. As decision-maker's attitude becomes
more risk-averse, seats are sold more to Fare 1 in earlier stage in time. However, as
decision-maker becomes less risk-averse, seats are kept unsold for longer period until
higher priced Fare 2 customer arrives. If decision maker wait until Fare 2 customer to
arrive, there is higher risk that he may not sell all seats by end of finite time horizon.
However, by selling seats to Fare 2 customer as many as possible, he can gain higher
wealth at the end. The more risk-averse decision maker's attitude is, less risk one
has to take and lower expected revenue and standard deviation. So, this simulation
result coincide with decision maker's risk attitude.
4.4
Network risk-averse model
It is natural to extend single leg model into muli-leg network risk-averse model. In
network risk-averse model, the airline company runs F fare products in L legged
network at the same time. We define the matrix A = [AI,f], where Ai,1 E {0, 1}
depending on whether fare
f
consumes a seat on leg I or not. Each fare product is
associated with a price P and requires seats on one or more legs. Initial capacity
on each leg is given by a vector xo E Zt. Time is discrete. We assume a T period
horizon with at most one customer arrival in a single period.
Similar to clustering model in Chapter 2, we can approximate the optimal value
function for network case by decomposing the network into individual legs instead of
clusters. Then associated reward for each fare product on particular leg I is Pf a,f
where fare coefficient a is defined as ai,f
We define the state-space S'
1,1
'
=
=
F1 Alf
{x
: 1 E Z+,X1 <
} x {w 1 : W E R+,w
> w1}
x
{t : t E Z+, t < T} and encoding
the products offered for sale at time t by a vector in a where a E
{O,
1}F
For each leg, 1, state transition function, S' is mapping S' x A S1-N S.
--.
It can be
described as below transition function.
(4.8)
(X' - A 1, 1 , W + Pa,
1,t
+ 1), w.p. Aia 1 IfX1>A, }
Sif 0 < t < T
(Xz, w1 , t + 1), w.p. 1-
(F Afaf{XI}> ,)
f=1
(9i, w t , t)
A control policy is a mapping 7r, : S 14 {0,
1}F.
,
otherwise
Let II, be the set of all such
policies. Let R'(x', w', t, a) be a random variable representing revenue generated by
the airline when fare products a E A are offered for sale, and we have x, seats left.
Then,
~T-1
(4.9)
J["(x', w', 0) = EE
R(X, 71 (X1) |X1 = xi
t=0
where X = S'((X _), 7rl(X1t_ )). Then we let maxr,
-nJ''(X,
w , t)
=
J1*(X1,
w
),
denote the expected revenue under the optimal policy 7r* upon starting in state
(Xi, w, t) E S.
J1* and 7r* can be computed via dynamic programming. We can define the dynamic
programming operator T according to
(T JI)(x', w', t) =
Af
-
J, (XI,w, t + 1)
f=1
F
(4.10)
+ E Af Max Ix>Al~} (Ji(x' - Al,f, w' + Pf a1 , I t + 1), Ji(iX, w1, t + 1))
f=1
af if
+ (I - I(;X1>Af}
J (. I, w1, t + 1))
We define (TJi)(xi, w t , T - 1) = u(wl) for the (T - 1)-th horizon. J* can then be
idenified as the unique solution to the fixed point equation TJ = J. Ir* is then the
policy that achieves the maximum utility in (4.10). So, r* = 0 iff Ji(x' - Ai, 1 , wl +
Pf aij, t + 1) < J, (xi, w, t+ 1) and t < T.
L
Sum of these leg wise decomposed value function,
E J,* (x', wl, t),
is an upper-
i=1
bound to optimal value function of the network, J* (x, w, t).
4.5
Network model numerical example
In this section, we examine numerical example of network risk-averse model. Example
network is depicted in Figure 4-3, and corresponding seat capacity and leg numbers
are listed in Table 4.2.
There are 18 fare products offered in this 10 leg network
within 1000 time steps. All corresponding fare product's price, customer arrival rate
and path are listed in Table 4.3.
All fare products' customer arrival rate is time
homogeneous, where high priced fare has low customer arrival rate and low priced
fare has high customer arrival rate.
Figure 4-3: Example network for network risk-averse model. Each legs are notated
with leg number and seat capacity in paranthesis.
Leg number
LI
L2
L3
L4
L5
L6
L7
L8
L9
L10
Start Node
1
1
1
2
2
3
3
5
4
3
End Node
2
3
6
3
4
2
4
3
6
5
Seat Capacity
150
50
50
100
200
100
100
50
150
50
Table 4.2: Expected revenue, standard deviation, and confidence interval of simulation result.
Fare Number
Price
Customer Arrival Rate
Origin
Destination
Path
1
2
3
4
5
6
160.89
216.85
266.44
224.58
328.55
160.92
0.068485844
0.023983763
0.025705747
0.015669475
1
1
1
1
2
3
4
5
1->2
1->3
1->2->4
1->6
7
8
9
10
135.04
271.67
183.21
124.87
11
12
13
14
1
6
1->3->5
0.074046935
0.003054652
0.024064375
0.057556518
2
2
2
2
3
3
4
5
6
2
2->3
2->4
2->3->5
2->4->6
3->2
131.64
0.050267934
3
4
156.71
154.8
0.016266819
0.001430706
3
5
3->4
3->5
3
15
73.88
288.75
0.058656346
0.002170956
4
5
6
6
2
3->4->6
4->6
5->3->2
16
17
18
222.21
288.85
298.45
0.015500202
0.000890830
0.000290694
5
5
5
3
4
6
5->3
5->3->4
5->3->4->6
0.005007962
0.056950242
Table 4.3: Expected revenue, standard deviation, and confidence interval of simulation result.
With Given setting, we can maximize expected utility by computing via dynamic
programming model from section 4.4. Similar to single leg risk-averse model, we used
u(w) = w(1-k) as our risk-averse utility function, where k is risk-aversion parameter.
Although the backward induction procedure for the finite horizon expected utility
objectives only considers the relevant wealth levels at the different decision time steps,
so the complexity of maintaining all wealth levels is too high. Suppose that the number of distinct non-zero fare price is n. In the worst case, the number of wealth levels
to be considered at time step t is
('
"), which is the number of combinations for se-
lecting t items from t+n classes allowing repetition. Thus, complexity of representing
different wealth level to perform backward induction is
(t")
=
n
= O(t"), an
n-th order polynomial in t, and thus computationally impractical if n is big. Therefore, some kind of approximation is needed and we uniformly discretized wealth levels
by increments of 50. We have chosen 50 as our wealth level increament, since 73.88
is the lowest price among fare products in our case.
Also, we discretized the wealth level into 1000 different wealth levels with increa-
ment of 50.
With different risk-aversion k values, we computed value functions via dynamic
programming and ran 100000 simulations from stored value function to calculate
expected utilities and standard deviations. Simulation results shows that As riskaversion parameter k increases, standard deviation and expected revenue decrease.
Risk Aversion
Parameter k
0
0.9
1-109
1-10-3
Expected
Revenue
99218.18718
99176.11992
99090.78858
98844.7545
Standard
Deviation
1384.517
1378.651
1365.587
1348.609
Confidence Interval
Range
Range
Minimum
Maximum
99213.809
99171.7602
99086.4702
98840.4898
99222.5654
99180.4796
99095.1069
98849.0192
Table 4.4: Expected revenue, standard deviation, and confidence interval of simulation result.
Figure 4-4: Expected revenue vs. standard deviation over different k value.
As shown in Figure 4-4, as risk-averse parameter k increases, standard deviation
decrease and also expected revenue decrease. As decision-maker's attitude becomes
more risk-averse, seats are sold more to high frequent and low priced fare in earlier
stage in time. However, as decision-maker becomes less risk-averse, seats are kept
unsold for longer period until high priced fares' customer arrives.
50
Chapter 5
Conclusions
We have presented effect of clustering in decomposed network revenue maximizing
capacity control model in a random environment and multi-leg network capacity
control problems from the perspective of a risk-averse decision-maker.
The first clustering model is an extension of existing capacity control model that
approximate the value function by decomposing the network into individual legs. By
clustering the legs, one can find tighter upperbound than leg-wise decomposition with
loss of computation speed due to larger state space. We have shown that there is more
than 6% revenue improvement opportunity by finding the right clustering. With local
interchange heuristic and generic heuristics, finding right clustering can be done in
faster time.
The second approach is advisable for companies that apply capacity control but
have a short time horizon, so that one single realization has a potential for severe
impact on the company revenues. Using different risk-aversion parameters on utility
function, decision maker can control not only the return of expected revenue but also
standard deviation of revenue that one prefers the most. We have shown that this riskaverse model not only works for single leg capacity control but it can be also applied
to network capacity control problem with leg wise decomposition approximation.
Unfortunately, the impact of accounting for risk-aversion is only best seen when the
fare arrival rates themselves are stochastic; in this case however, even the risk neutral
problem is difficult to solve.
5.1
Directions for future research
There are several directions for future work.
First, there is revenue improvement opportunity if we find optimal fare coefficient
for decomposition approximation that we did for network problem on both risk-averse
and risk-neutral model. In our model, we divided reward of fare product to corresponding legs evenly. However, depending on operation cost for each leg, one can
optimize further more to maximize the revenue of the company.
Second, there is also precision improvement opportunity if we can use piecewise
linear approximation or cubic spline approximation for discretizing wealth levels in
risk-averse model. In our case, we discretized the wealth level in equi-distant apart
to each other. However, with more advanced method like piecewise linear approximation, one can achieve tighter error bound on approximating the revenue and standard
deviation calculation on risk-averse model.
Finally, one can consider cancelations or no-shows for the models that we proposed. No-shows and cancelations are very common phenomena in practice, so it
would be very interesting topic for future research.
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